I like Wiebel immensely as well. But the most readable introduction I've seen to the topic is Bott and Tu's classic DIFFERENTIAL FORMS IN ALGEBRAIC TOPOLOGY. You can also try the nice presentation in the second edition of Joseph Rotman's homological algebra book.That should help you,Colin.
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The MathemagicianApr 22 '10 at 17:40

16 Answers
16

Many of the references that people have mentioned are very nice, but the brutal truth
is that you have to work very hard through some basic examples before it really makes
sense.

Take a complex $K=K^\bullet$ with a two step filtration $F^1\subset F^0=K$, the spectral
sequence contains no more information than is contained in the long exact sequence associated
to
$$0 \to F^1\to F^0\to (F^1/F^0)\to 0$$
Now consider a three step filtration $F^2\subset F^1\subset F^0=K$, write down all the short
exact sequences you can and see what you get. The game is to somehow relate $H^*(K)$
to $H^*(F^i/F^{i+1})$. Suppose you know these are zero, is $H^*(K)=0$? Once you've mastered
that then ...

@Donu: I fixed it. As you can see, when the latex isn't displaying right for no apparent reason, you should try enclosing the expression in backticks, i.e., `$ \$ ... \$ $'.
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Pete L. ClarkApr 22 '10 at 19:46

Can't upvote this enough. The best way to come to grips with spectral sequences is to get your hands unrecognizably dirty with lots of examples and manual computation.
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Eric PetersonApr 23 '10 at 1:20

Bott and Tu, "Differential forms in Algebraic Topology" has some very nice exposition on spectral sequences. It has a fairly geometrical starting point, motivating the whole subject by generalizing the Meyer-Vietoris sequence to more complicated coverings and relating Cech cohomology to de Rham cohomology.

I'm pleased to see Bott-Tu recommended; it's one of my favorite books. When I was a graduate student (that's thirty years ago!), the best, by far, exposition of spectral sequences was in Bott's course. This became part of the book by Bott and Tu, with, I believe, some help from Dan Freed who took Bott's course as an undergraduate. But since I'm far from an expert and have not read anything about spectral sequences since then, I was not sure whether something clearly better had appeared by now.
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Deane YangApr 22 '10 at 15:56

1

As you say, the motivation is important. I've recommended the book to people, only for them to jump to chapter 3, titled spectral sequences, but chapter 2 is already on the topic.
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Ben WielandApr 24 '10 at 22:33

The book which cured my fear of spectral sequences is "Cohomology Operations and Applications in Homotopy Theory" by Moser and Tangora. It only touches applications in topology, and by todays standards it would be considered very basic; the upside of this is that a lot of the material is passed in the exercises (another upside is that it's $10 on Amazon).

I liked Mosher and Tangora quite a lot, too. An advantage of its concreteness and focus on specific applications is that there are lots of calculations which they do and which you, too, can do explicitly-- this is one of the subjects where it is hard to have a solid understanding without doing a fair number of examples and computations oneself.
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Allison SmithApr 22 '10 at 16:36

I think the book is really pretty good at introducing spectral sequences because, like hatcher, it has an application/computation in mind that you can do right away.
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Sean TilsonMay 20 '10 at 5:00

I would recommend that everyone's very first (zeroth?) introduction would be Timothy Chow's excellent short article You Could Have Invented Spectral Sequences. It doesn't give a lot of technical details, but it will definitely remove your fear before you start on a more advanced exposition.

Ken Brown's book, "Cohomology of groups" also gives a fairly readable introduction to spectral sequences. The algebra is kept fairly simple here, and most of the discussion is about computing the homology of a double complex, and constructing the Lyndon-Hochschild-Serre spectral sequence. So it doesn't go particularly deep, but it's a non-frightening place to start.

I'm going to have to agree with everyone who recommends Bott & Tu. That provided me with a good understanding of the basic setup. After I was comfortable with that, I moved on to Hilton & Stammbach's book "A Course in Homological Algebra" that did a good job of showing how the general idea works for Abelian categories.